Having found that $f\,''(x) = 0$ for some value of $x$, one can evaluate successive higher derivatives at the same value of $x$ until one finds a higher derivative that is non-zero there. If that's an odd-order derivative, there is a point of inflection there.

If the function has all its higher derivatives equal to zero, the issue still isn't decided.

There are also functions such as $f(x) = x^{1/3}$ that have a tangent that crosses the curve at its point of contact with the curve, but aren't differentiable there.